A note on semi-discrete modelling in the life sciences

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 367, Issue 1908, 2009, Pages 4779-4799

  Mailleret, Ludovic ^a   Lemesle, Valérie ^b,c  

^a CEMAGREF   (France)

^b UNITÉ DE MATHÉMATIQUES PURES ET APPLIQUÉES   (France)

^c CIRAD   (France)

Author keywords

Allee effect model; Beverton Holt model; Epidemiology; Impulsive differential equations; Medicine; Population dynamics

Indexed keywords

CONTINUOUS TIME SYSTEMS; DIFFERENTIAL EQUATIONS; DYNAMICAL SYSTEMS; POPULATION DYNAMICS;

ALLEE EFFECT MODEL; BEVERTON-HOLT MODEL; EFFECT MODEL; IMPULSIVE DIFFERENTIAL EQUATION; IMPULSIVE DIFFERENTIAL EQUATIONS;

DYNAMICS;

ANIMAL; BIOLOGICAL MODEL; FISH; GROWTH, DEVELOPMENT AND AGING; POPULATION DYNAMICS; REVIEW; STATISTICAL MODEL;

ANIMALS; FISHES; MODELS, BIOLOGICAL; MODELS, STATISTICAL; POPULATION DYNAMICS;

EID: 73349121278     PISSN: 1364503X     EISSN: None     Source Type: Journal    
DOI: 10.1098/rsta.2009.0153     Document Type: Article

References (104)

1. Agur, Z., Cojocaru, L., Mazor, G., Anderson, R. M. & Danon, Y. L. 1993 Pulse mass measles vaccination across age cohorts. Proc. Natl Acad. Sci. USA 90, 11698-11702. (doi:10.1073/nas.90.24.11698)
2. Akhmet, M., Beklioglu, M., Ergenc, T. & Tkachenko, V. 2006 An impulsive ratio-dependent predator-prey system with diffusion. Nonlinear Anal.: Real World Appl. 7, 1255-1267. (doi:10.1016/j.nonrwa.2005.11. 007)
3. Akhmetzhanov, A., Grognard, F. & Mailleret, L. In preparation. On the evolution of a foraging reproduction trade off and its consequences on consumer-resource dynamics.
4. Andreasen, V. & Frommelt, T. 2005 A school-oriented, age-structured epidemic model. SIAM J. Appl. Math. 65, 1870-1887. (doi:10.1137/040610684)
5. Armstrong, D. P. & Seddon, P. J. 2008 Directions in reintroduction biology. Trends Ecol. Evol. 23, 20-25. (doi:10.1016/j.tree.2007. 10.003)
6. Bainov, D. D. & Simeonov, P. S. 1989 Systems with impulsive effect: stability theory and applications. Chichester, UK: Ellis Horwood.
7. Ballinger, G. & Liu, X. 1997 Permanence of population growth models with impulsive effects. Math. Comput. Model. 26, 59-72. (doi:10.1016/S0895-7177(97)00240-9)
8. Beverton, R. J. H. & Holt, S. J. 1957 On the dynamics of exploited fish populations. In Fisheries investigations, series 2, vol. 19. London, UK: Her Majesty's Stationery Office.
9. Braverman, E. & Mamdani, R. 2008 Continuous versus pulse harvesting for population models in constant and variable environment. J. Math. Biol. 57, 413-434. (doi:10.1007/s00285-008-0169-z)
10. Briggs, C. J. & Godfray, H. C. J. 1996 The dynamics of insect-pathogen interactions in seasonal environments. Theor. Popul. Biol. 50, 149-177. (doi:10.1006/tpbi.1996.0027)
11. Cappuccio, A., Castiglione, F. & Piccoli, B. 2007 Determination of the optimal therapeutic protocols in cancer immunotherapy. Math. Biosci. 209, 1-13. (doi:10.1016/j.mbs.2007.02.009)
12. Choisy, M., Guegan, J. F. & Rohani, P. 2006 Dynamics of infectious diseases and pulse vaccination: teasing apart the embedded resonance effects. Physica D 223, 26-35. (doi:10.1016/j.physd.2006.08.006)
13. Clark, C. W. 1990 Mathematical bioeconomics: the optimal management of renewable resources, 2nd edn. New York, NY: John Wiley & Sons, Inc.
14. Courchamp, F., Clutton-Brock, T. & Grenfell, B. 1999 Inverse density dependence and the Allee effect. Trends Ecol. Evol. 14, 405-410. (doi:10.1016/S0169-5347(99)01683-3)
15. Deredec, A. & Courchamp, F. 2007 Importance of the Allee effect for reintroductions. Ecoscience 14, 440-451. (doi:10.2980/1195- 6860(2007)14[440:IoTAEF]2.0.Co;2)
16. de Roos, A. M., Diekmann, O. & Metz, J. A. J. 1992 Studying the dynamics of structured population models: a versatile technique and its application to Daphnia. Am. Nat. 139, 123-147. (doi:10.1086/285316)
17. Ding, W. 2007 Optimal control on hybrid ODE systems with application to a tick disease model. Math. Biosci. Eng. 4, 633-659.
18. Dong, L., Chen, L. & Sun, L. 2007 Optimal harvesting policies for periodic Gompertz systems. Nonlinear Anal.: Real World Appl. 8, 572-578. (doi:10.1016/j.nonrwa.2006.01.001)
19. D'Onofrio, A. 2002 Stability properties of pulse vaccination strategy in SEIR epidemic model. Math. Biosci. 179, 57-72. (doi:10.1016/S0025-5564(02)00095-0)
20. D'Onofrio, A. 2005 On pulse vaccination strategy in the SIR epidemic model with vertical transmission. Appl. Math. Lett. 18, 729-732. (doi:10.1016/j.aml.2004.05.012)
21. Drury, K. L. S. & Lodge, D. M. 2008 Using mean first passage times to quantify equilibrium resilience in perturbed intraguild predation systems. Theor. Ecol. 2, 41-51. (doi:10.1007/s12080-008-0027-z)
22. Ebenhoh, W. 1988 Coexistence of an unlimited number of algal species in a model system. Theor. Popul. Biol. 34, 130-144. (doi:10.1016/0040-5809(88)90038-X)
23. Eskola, H. T. M. & Geritz, S. A. H. 2007 On the mechanistic derivation of various discrete-time population models. B. Math. Biol. 69, 329-346. (doi:10.1007/s11538-006-9126-4)
24. Eskola, H. T. M. & Parvinen, K. 2007 On the mechanistic underpinning of discrete- time population models with Allee effect. Theor. Popul. Biol. 72, 4l-5l. (doi:10.1016/ j.tpb.2007.03.004)
25. Fenton, A., Norman, R., Fairbairn, J. P. & Hudson, P. J. 2001 Evaluating the efficacy of entomopathogenic nematodes for the biological control of crop pests: a nonequilibrium approach. Am. Nat. 158, 408-425. (doi:10.1086/321993)
26. Fuhrman, K. M., Lauko, I. G. & Pinter, G. A. 2004 Asymptotic behavior of an SI epidemic model with pulse removal. Math. Comput. Model. 40, 371-386. (doi:10.1016/j.mcm.2003.10.047)
27. Funasaki, E. & Kot, M. 1993 Invasion and chaos in a periodically pulsed mass-action chemostat. Theor. Popul. Biol. 44, 203-224. (doi:10.1006/tpbi.1993.1026)
28. Gakkhar, S. & Negi, K. 2008 Pulse vaccination in SIRS epidemic model with non-monotonic incidence rate. Chaos Solitons Fractals 35, 626-638. (doi:10.1016/j.chaos.2006.05.054)
29. Galvani, A. P., Reluga, T. C. & Chapman, G. B. 2007 Long-standing influenza vaccination policy is in accord with individual self-interest but not with the utilitarian optimum. Proc. Natl Acad. Sci. USA 104, 5692-5697.
30. Gao, S. J., Chen, L. S. & Sun, L. H. 2005 Dynamic complexities in a seasonal prevention epidemic model with birth pulses. Chaos Solitons Fractals 26, 1171-1181. (doi:10.1016/j.chaos.2005.02.032)
31. Gao, S. J., Chen, L. S., Nieto, J. J. & Torres, A. 2006 Analysis of a delayed epidemic model with pulse vaccination and saturation incidence. Vaccine 24, 6037-6045. (doi:10.1016/j.vaccine.2006.05.018)
32. Gao, S. J., Chen, L. S. & Teng, Z. D. 2007 Impulsive vaccination of an SEIRS model with time delay and varying total population size. B. Math. Biol. 69, 731-745. (doi:10.1007/s11538-006-9149-x)
33. Georgescu, P. & Morosanu, G. 2008 Impulsive perturbations of a three-trophic prey-dependent food chain system. Math. Comput. Model. 48, 975-997. (doi:10.1016/j.mcm.2007.12.006)
34. Geritz, S. A. H. & Kisdi, E. 2004 On the mechanistic underpinning of discrete-time population models with complex dynamics. J. Theor. Biol. 228, 261-269. (doi:10.1016/j.jtbi.2004.01.003)
35. Ghosh, M. & Pugliese, A. 2004 Seasonal population dynamics of ticks, and its influence on infection transmission: a semi-discrete approach. B. Math. Biol. 66, 1659-1684. (doi:10.1016/j.bulm.2004.03.007)
36. Gubbins, S. & Gilligan, C. A. 1997a Biological control in a disturbed environment. Phil. Trans. R. Soc. Lond. B 352, 1935-1949. (doi:10.1098/rstb.1997.0l80)
37. Gubbins, S. & Gilligan, C. A. 1997b Persistence of host-parasite interactions in a disturbed environment. J. Theor. Biol. 188, 241-258. (doi:10.1006/jtbi.1997.0466)
38. Gyllenberg, M., Hanski, I. & Lindstrom, T. 1997 Continuous versus discrete single species population models with adjustable reproductive strategies. B. Math. Biol. 59, 679-705. (doi:10.1007/BF02458425)
39. Hall, R. J., Gubbins, S. & Gilligan, C. A. 2007 Evaluating the performance of chemical control in the presence of resistant pathogens. B. Math. Biol. 69, 525-537. (doi:10.1007/s11538-006-9139-z)
40. Hochberg, M. E., Hassell, M. P. & May, R. M., 1990 The dynamics of host-parasitoid-pathogen interactions. Am. Nat. 135, 74-94. (doi:10.1086/285033)
41. Ives, A. R., Gross, K. & Jansen, V. A. A. 2000 Periodic mortality events in predator-prey systems. Ecology 81, 3330-3340.
42. Iwasa, Y. & Cohen, D. 1989 Optimal growth schedule of a perennial plant. Am. Nat. 133, 480-505. (doi:10.1086/284931)
43. Jansen, V. A. A. & Sabelis, M. W. 1995 Outbreaks of colony-forming pests in tri-trophic systems - consequences for pest-control and the evolution of pesticide resistance. Oikos 74, 172-176. (doi:10.2307/3545688)
44. Johnson, D. M., Liebhold, A. M., Tobin, P. C. & Bjornstad, O. N. 2006 Allee effects and pulsed invasion by the gypsy moth. Nature 444, 361-363. (doi:10.1038/nature05242)
45. Keitt, T. H., Lewis, M. A. & Holt, R. D. 2001 Allee effects, invasion pinning, and species' borders. Am. Nat. 157, 203-216. (doi:10.1086/318633)
46. Kisdi, E. & Utz, M. 2005 Does quasi-local competition lead to pattern formation in metapopulations? An explicit resource competition model. Theor. Popul. Biol. 68, 133-145. (doi:10.1016/j.tpb.2005.06.001)
47. Kroll, M. H. 2000 Parathyroid hormone temporal effects on bone formation and resorption. B. Math. Biol. 62, 163-188. (doi:10.1006/bulm. 1999.0146)
48. Lakmeche, A. & Arino, O. 2000 Bifurcation of non-trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment. Dyn. Contin. Discret. Impul. Syst. 7, 265-287.
49. Lakmeche, A. & Arino, O. 2001 Nonlinear mathematical model of pulsed-therapy of heterogeneous tumors. Nonlinear Anal.: Real World Appl. 2, 455-465.
50. Li, W. T. & Fan, Y. H. 2007 Existence and global attractivity of positive periodic solutions for the impulsive delay Nicholson's blowflies model. J. Comput. Appl. Math. 201, 55-68. (doi:10.1016/j.cam.2006.02. 001)
51. Li, Z. Q., Wang, W. M. & Wang, H. L. 2006 The dynamics of a Beddington-type system with impulsive control strategy. Chaos Solitons Fractals 29, 1229-1239. (doi:10.1016/j.chaos.2005.08.195)
52. Liu, K. Y. & Chen, L. S. 2008 On a periodic time-dependent model of population dynamics with stage structure and impulsive effects. Discrete Dyn. Nat. Soc. 389727. (doi:10.1155/2008/389727)
53. Liu, M. X. & Jin, Z. 2006 The effect of impulsive control strategy in HIV/AIDS model. Dyn. Cont. Disc. Ser. A 13, 706-713.
54. Liu, B., Chen, L. S. & Zhang, Y. J. 2005 The dynamics of a prey-dependent consumption model concerning impulsive control strategy. Appl. Math. Comput. 169, 305-320. (doi:10.1016/j.amc.2004.09.053)
55. Liu, H. L., Yu, J. Y. & Zhu, G. T. 2008a Global behaviour of an age-infection-structured HIV model with impulsive drug-treatment strategy. J. Theor. Biol. 253, 749-754. (doi:10.1016/j.jtbi.2008.04.021)
56. Liu, X. N., Takeuchi, Y. & Iwami, S. 2008b SVIR epidemic models with vaccination strategies. J. Theor. Biol. 253, 1-11. (doi:10.1016/j.jtbi.2007.10.014)
57. Lu, Z. H., Chi, X. B. & Chen, L. S. 2003 Impulsive control strategies in biological control of pesticide. Theor. Popul. Biol. 64, 39-47. (doi:10.1016/S0040-5809(03)00048-0)
58. Madden, L. V. & van den Bosch, F. 2002 A population-dynamics approach to assess the threat of plant pathogens as biological weapons against annual crops. Bioscience 52, 65-74. (doi:10.1641/0006-3568(2002)052[0065: APDATA]2.0.Co;2)
59. Maggi, S. & Rinaldi, S. 2006 A second-order impact model for forest fire regimes. Theor. Popul. Biol. 70, 174-182. (doi:10.1016/j.tpb.2006.01.007)
60. Mailleret, L. & Grognard, F. 2006 Optimal release policy for prophylactic biological control. In Positive systems (eds C. Commault & N. Marchand). Lecture Notes in Control of Information, vol. 341, pp. 89-96. Berlin, Germany: Springer.
61. Mailleret, L. & Grognard, F. 2009 Global stability and optimisation of a general impulsive biological control model. Math. Biosci. 221, 91-100. (doi:10.1016/j.mbs.2009.07.002)
62. Malthus, T. 1798 An essay on the principle of population. Harmondsworth.
63. Meng, X. Z. & Chen, L. S. 2008 The dynamics of a new SIR epidemic model concerning pulse vaccination strategy. Appl. Math. Comput. 197, 582-597. (doi:10.1016/j.amc.2007.07.083)
64. Murdoch, W. W., Briggs, C. & Nisbet, R. M. 2003 Consumer resource dynamics. Princeton, NJ: Princeton University Press.
65. Nedorezov, L. V. & Nedorezova, B. N. 1995 Correlation between models of population-dynamics in continuous and discrete-time. Ecol. Model. 82, 93-97. (doi:10.1016/0304-3800(94)00069-T)
66. Nicholson, A. & Bailey, V. A. 1935 The balance of animal populations. Proceedings Zoological Society London Part 1 3, 551-598.
67. Nundloll, S., Mailleret, L. & Grognard, F. 2008 The effect of partial crop harvest on biological pest control. Rocky Mt. J. Math. 38, 1633-1661. (doi:10.1216/RMJ-2008-38-5-1633)
68. Nundloll, S., Mailleret, L. & Grognard, F. In press. Two models of interfering predators in impulsive biological control. J. Biol. Dyn.(doi:10.1080/17513750902968779)
69. Nundloll, S., Mailleret, L. & Grognard, F. Submitted. Impulsive biological control with intrapredatory interference: global stability and convergence.
70. Pachepsky, E., Nisbet, R. M. & Murdoch, W. W. 2008 Between discrete and continuous: consumer-resource dynamics with synchronized reproduction. Ecology 89, 280-288. (doi:10.1890/07-0641.1)
71. Panetta, J. C. 1996 A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competitive environment. B. Math. Biol. 58, 425-447. (doi:10.1007/BF02460591)
72. Panetta, J. C. 1998 A mathematical model of drug resistance: heterogeneous tumors. Math. Biosci. 147, 41-61. (doi:10.1016/S0025-5564(97)00080-1)
73. Persson, L. & de Roos, A. M. 2003 Adaptive habitat use in size-structured populations: linking individual behavior to population processes. Ecology 84, 1129-1139. (doi:10.1890/0012-9658(2003) 084[1129:AHUISP]2.0.co;2)
74. Pietravalle, S., Lemarie, S. & van den Bosch, F. 2006 Durability of resistance and cost of virulence. Eur. J. Plant Pathol. 114, 107-116. (doi:10.1007/s10658-005-3479-7)
75. Roberts, M. G. & Kao, R. R. 1998 The dynamics of an infectious disease in a population with birth pulses. Math. Biosci. 149, 23-36. (doi:10.1016/S0025-5564(97)10016-5)
76. Rueffler, C., Van Dooren, T. J. M. & Metz, J. A. J. 2006 The evolution of resource specialization through frequency-dependent and frequency-independent mechanisms. Am. Nat. 167, 81-93. (doi:10.1086/498275)
77. Rueffler, C., Van Dooren, T. J. M. & Metz, J. A. J. 2007 The interplay between behavior and morphology in the evolutionary dynamics of resource specialization. Am. Nat. 169, E34-E52. (doi:10.1086/510635)
78. Schaffer, W. M. 1983 The application of optimal control theory to the general life history problem. Am. Nat. 121, 418-431. (doi:10.1086/284070)
79. Sharov, A. A. & Liebhold, A. M. 1998 Model of slowing the spread of the gypsy moth (lepidoptera: Lymantriidae) with a barrier zone. Ecol. Appl. 8, 1170-1179. (doi:10.1890/1051-0761(1998)008[1170:MOSTSO]2.0. Co;2)
80. Shaw, M. W. 1994 Seasonally induced chaotic dynamics and their implications in models of plant-disease. Plant Pathol. 43, 790-801. (doi:10.1111/j.1365-3059.1994.tb01623.x)
81. Sherratt, J. A., Eagan, B. T. & Lewis, M. A. 1997 Oscillations and chaos behind predator-prey invasion: mathematical artifact or ecological reality? Phil. Trans. R. Soc. Lond. B 352, 21-38. (doi:10.1098/rstb.1997.0003)
82. Shi, R. Q. & Chen, L. S. 2008 Staged-structured Lotka-Volterra predator-prey models for pest management. Appl. Math. Comput. 203, 258-265. (doi:10.1016/j.amc.2008.04.032)
83. Shulgin, B., Stone, L. & Agur, Z. 1998 Pulse vaccination strategy in the SIR epidemic model. B. Math. Biol. 60, 1 123-1 148. (doi:10.1016/S0092-8240(98)90005-2)
84. Shurilov, A., Medvedev, A. & Shepeljavyi, A. 2009 Mathematical model of non-basal testosterone regulation in the male by pulse modulated feedback. Automatica 45, 78-85. (doi:10.1016/j.automatica.2008.06.016)
85. Singh, A. & Nisbet, R. M. 2007 Semi-discrete host-parasitoid models. J. Theor. Biol. 247, 733-742. (doi:10.1016/j.jtbi.2007.04.004)
86. Smith?, R. J. & Schwartz, E. J. 2008 Predicting the potential impact of a cytotoxic T-lymphocyte HIV vaccine: how often should you vaccinate and how strong should the vaccine be? Math. Biosci. 212, 180-187. (doi:10.1016/j.mbs.2008.02.001)
87. Song, X. Y. & Zhao, Z. 2006 Extinction and permanence of two-nutrient and one- microorganism chemostat model with pulsed input. Discrete Dyn. Nat. Soc. 31614. (doi:10.1155/DDNS/2006/31614)
88. Swinton, J., Tuyttens, F., Macdonald, D., Nokes, D. J., Cheeseman, C. L. & Clifton-hadley, R. 1997 Comparison of fertility control and lethal control of bovine tuberculosis in badgers: the impact of perturbation induced transmission. Phil. Trans. R. Soc. Lond. B 352, 619-631. (doi:10.1098/rstb.1997.0042)
89. Tang, S. Y. & Chen, L. S. 2004 The effect of seasonal harvesting on stage-structured population models. J. Math. Biol. 48, 357-374. (doi:10.1007/s00285-003-0243-5)
90. Taylor, C. M. & Hastings, A. 2005 Allee effects in biological invasions. Ecol. Lett. 8, 895-908. (doi:10.1111/j.l46l-0248.2005. 00787.x)
91. Toth, D. J. A. 2008 Strong resonance and chaos in a single-species chemostat model with periodic pulsing of resource. Chaos Solitons Fractals 38, 55-69. (doi:10.1016/j.chaos.2006.12.001)
92. Truscott, J. E., Webb, C. R. & Gilligan, C. A. 1997 Asymptotic analysis of an epidemic model with primary and secondary infection. B. Math. Biol. 59, 1101-1123. (doi:10.1007/BF02460103)
93. Utz, M.' Kisdi, E. & Doebeli, M. 2007 Quasi-local competition in stage-structured metapopulations: a new mechanism of pattern formation. B. Math. Biol. 69, 1649-1672. (doi:10.1007/s11538-006-9184-7)
94. van de Wolfshaar, K. E., de Roos, A. M. & Persson, L. 2008 Population feedback after successful invasion leads to ecological suicide in seasonal environments. Ecology 89, 259-268. (doi:10.1890/06-2058.1)
95. Vendite, L. L. & Ghini, R. 1999 A mathematical model for fungal population growth and the fungicide resistance problem. J. Biol. Syst. 7, 239-249. (doi:10.1142/S0218339099000164)
96. Verhulst, P. 1838 Notices sur la loi que la population suit dans son accroissement. Correspondance mathématique et physique 10, 113-121.
97. Vidal, A., Médigue, C., Malpaux, B. & Clément, F. 2009 Endogenous circannual rhythm in luteinizing hormone secretion: insight from signal analysis coupled with mathematical modelling. Phil. Trans. R. Soc. A 367, 4759-4777. (doi:10.1098/rsta.2009.0136)
98. Wagner, B. G. & Earn, D. J. D. 2008 Circulating vaccine derived polio viruses and their impact on global polio eradication. B. Math. Biol. 70, 253-280. (doi:10.1007/s11538-007-9255-4)
99. Wang, F. Y., Hao, C. P. & Chen, L. S. 2007 Bifurcation and chaos in a Monod type food chain chemostat with pulsed input and washout. Chaos Solitons Fractals 31, 826-839. (doi:10.1016/j.chaos.2005.10.044)
100. Wang, S. C., Zhang, B. Y., Li, Z. Q., Cressman, R. & Tao, Y. 2008 Evolutionary game dynamics with impulsive effects. J. Theor. Biol. 254, 384-389. (doi:10.1016/j.jtbi.2008.05.016)
101. Xiang, Z. G., Song, X. Y. & Zhang, F. Q. 2008 Bifurcation and complex dynamics of a two-prey two-predator system concerning periodic biological and chemical control. Chaos Solitons Fractals 37, 424-437. (doi:10.1016/j.chaos.2006.09.024)
102. Xiao, Y., Cheng, D. & Qin, H. 2006 Optimal impulsive control in periodic ecosystem. Syst. Control Lett. 55, 558-565. (doi:10.1016/j.sysconle.2005.12.003)
103. Zhang, X. S. & Holt, J. 2001 Mathematical models of cross protection in the epidemiology of plant-virus diseases. Phytopathology 91, 924-934. (doi:10.l094/PHYTO.2001.91.10.924)
104. Zhang, X., Shuai, Z. & Wang, K. 2003 Optimal impulsive harvesting policy for single population. Nonlinear Anal: Real World Appl. 4, 639-651. (doi:10.1016/S1468-1218(02)00084-6)